Question

Please round all entries below to the nearest ten-thousandth, i.e. nearest 4th decimal place, where necessary. You are salling your boat far off the west coast of some continent. There are three radio stations on the coast, call them stations A, B, and C, which conveniently all lie on a signle line! Your map tells you that station A is 500 miles north of station B, and station B is 1500 miles north of station C. Let's say that station B is located at the point (0, 0), and that the directions north and east are positive, while south and west are negative. With all this in mind... What are the coordinates of port A? What are the coordinates of port C? Using your onboard computer radio, which sends and receives signals to and from the radio stations and marks the response times, you determine that you are 3860.052 miles from station A, 4272.002 miles from station B, and 5594.64 miles from station C You recall your algebra class from long ago, and realize that your boat is at some point (, ) that lies on two hyperbolas: #2: ㄧㄏㄒ-1 Here hyperbola #1 is the one whose focal points are at stations A and B. Hyperbola #2 has focal points at stations B and C. You can solve this system for (, y) once you have determined the parameters, a1, bi, n2, a2,b... With those parameters, what is the solution (x, y) to the above system?i.e. where is your boat located? Note: This system can be solved by hand without too much trouble. Some careful substitution and quadratic formulai should do it. I'll give you additional extra credit if you show me the details on paper. If you're in a hurry, you can just as well solve this using a graphing utlity. Check out this template at DESMOS LORAN template colinear focl Here you can specify parameters ni, a1, bi, ng, a, b directly, as well as the locations of the ports. Where do the resulting hyperbolas intersect? There many be several solutions. Which one is the solution that marks the location of your boat?

Answer 1

NOTE : I have solved the problem for extra credits. Have used valid approximations where required. Please go through the below solution and let me know in case of any doubts.

Sol 02 5cn) lw 0 jo) S 00 - 1500 Rupe' bola #1 2. (0)の from the questirm, we Kono di dy 7a miles 2 32/02- +52,-- 2스-18432 pes bola # al= 135-72, 136 32 9216

12502 43758 2 hafa, bola#} Putin 18432 216 143 758 2 3けら2]) _ 181 32 7582 2 t 7 50) 32175 1.8132 6 589110d 955, 4 6 8 800 152 7忤6331504-131S:A g425 二 IS27y +653/S0y 11370 6722S 13 7582 sR5 comes mejahve Se Not Possi ble So 615 6 75ヤ75e7-43158 2 | 125023 31582 15130"13x 1503 455153.6 37582 x,'3 = 675, 615